Introduction to Classical Mechanics · Problem 4.23

Conservation of energy and momentum — Morin

Physics — Mechanics · pendulum-corrections

Problem

(a) For small oscillations, the period of a pendulum is approximately T≈2πℓ/gT \approx 2\pi\sqrt{\ell/g}, independent of the amplitude, θ0\theta_0. For finite oscillations, use dt=dx/vdt = dx/v to show that the exact expression for TT is T=8ℓg∫0θ0dθcos⁡θ−cos⁡θ0.T = \sqrt{\frac{8\ell}{g}}\int_0^{\theta_0} \frac{d\theta}{\sqrt{\cos\theta - \cos\theta_0}}. (b) Find an approximation to this TT, up to second order in θ02\theta_0^2, in the following way. Make use of the identity cos⁡ϕ=1−2sin⁡2(ϕ/2)\cos\phi = 1 - 2\sin^2(\phi/2) to write TT in terms of sines (because it's more convenient to work with quantities that go to zero as θ→0\theta \to 0). Then make the change of variables, sin⁡x≡sin⁡(θ/2)/sin⁡(θ0/2)\sin x \equiv \sin(\theta/2)/\sin(\theta_0/2) (you'll see why). Finally, expand your integrand in powers of θ0\theta_0, and perform the integrals to show that T≈2πℓg(1+θ0216+⋯ ).T \approx 2\pi\sqrt{\frac{\ell}{g}}\left(1 + \frac{\theta_0^2}{16} + \cdots\right).

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